for your first question: first imagine there is a largest prime number, X.
Think about how you could prove there is a larger one.
hint: think about factorials
1. Prove that there is no largest prime.
Proof: Assume by contradiction that there is a largest prime . Then its divisors are and . Then how would I continue?
2. If is a positive integer, prove that the algebraic identity . So use induction on ?
3. If is prime, prove that is prime. Then is not prime. But this does not imply that is prime.